#148197

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,4)(5,6)(7,8)(9,10,15,21,28,37,29,17,12,18,13,19,14,20,30,36)(11,22,27,32,26,34,38,40,24,16,25,31,23,33,39,35);(1,2,3,4)(9,11,12,24)(10,16,18,22)(13,25,15,27)(14,23,28,26)(17,35,36,40)(19,32,21,31)(20,34,37,33)(29,38,30,39);(6,8)(10,17)(11,23)(13,29)(14,28)(15,30)(16,31)(18,36)(19,20)(21,37)(22,32)(24,26)(25,27)(33,40)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55393);